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A key ring has 7 keys. How many different ways can the keys be arrange

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A key ring has 7 keys. How many different ways can the keys be arrange [#permalink]

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Re: A key ring has 7 keys. How many different ways can the keys be arrange [#permalink]

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Bunuel wrote:
A key ring has 7 keys. How many different ways can the keys be arranged?

A. 6
B. 7
C. 120
D. 720
E. 5040


The thing about a key ring is that, since the keys can spin around the ring, the positions of the keys are all relative to each other.
So, let's take the task of arranging the keys and break it into stages.

Let's call the 7 keys A, B, C, D, E, F and G

Stage 1: Place the A key on the ring.
We don't care about the "position" of the ring because all positions are the same. In other words, if I asked you "In how many ways can we place 1 key on a key ring?" the answer is 1.
So, we can complete stage 1 in 1 way.

Stage 2: Select a key to go to the immediate right of key A (i.e, clockwise from key A)
There are 6 remaining keys to digits from which to choose, so we can complete this stage in 6 ways.

Stage 3: Select a key to go to the immediate right of the key we selected in stage 2.
There are 5 keys remaining, so we can complete this stage in 5 ways.

Stage 4: Select a key to go to the immediate right of the key we selected in stage 3.
There are 4 keys remaining, so we can complete this stage in 4 ways.

Stage 5: Select a key to go to the immediate right of the key we selected in stage 4.
There are 3 keys remaining, so we can complete this stage in 3 ways.

Stage 6: Select a key to go to the immediate right of the key we selected in stage 5.
There are 2 keys remaining, so we can complete this stage in 2 ways.

Stage 7: Select a key to go to the immediate right of the key we selected in stage 6.
There is 1 key remaining, so we can complete this stage in 1 way.

By the Fundamental Counting Principle (FCP), we can complete all 7 stages (and thus arrange all 7 keys) in (1)(6)(5)(4)(3)(2)(1) ways (= 720 ways)

Answer: C

Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT, so be sure to learn this technique.

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Fundamental Counting Principle (FCP)



Fundamental Counting Principle - example

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Re: A key ring has 7 keys. How many different ways can the keys be arrange [#permalink]

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New post 23 Feb 2017, 19:56
Arranging keys in a ring means arranging keys in a circular pattern

so arranging 7 keys can be done in (7-1)! = 6! = 720

Option D
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A key ring has 7 keys. How many different ways can the keys be arrange [#permalink]

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New post 23 Feb 2017, 20:02
Bunuel wrote:
A key ring has 7 keys. How many different ways can the keys be arranged?

A. 6
B. 7
C. 120
D. 720
E. 5040


This situation would be called a "circular permutation" and there are two cases of circular-permutations:-
(a)If clockwise and anti clock-wise orders are different, then total number of circular-permutations is given by (n-1)!
(b)If clock-wise and anti-clock-wise orders are taken as not different, then total number of circular-permutations is given by (n-1)!/2!

Since this is a case where clockwise and anti clock-wise orders are different, the formula is (n-1)!
n=7
(n-1)! = 6! =720

Hence Option D is correct.
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Re: A key ring has 7 keys. How many different ways can the keys be arrange [#permalink]

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New post 23 Feb 2017, 20:15
GMATPrepNow wrote:
Bunuel wrote:
A key ring has 7 keys. How many different ways can the keys be arranged?

A. 6
B. 7
C. 120
D. 720
E. 5040


The thing about a key ring is that, since the keys can spin around the ring, the positions of the keys are all relative to each other.
So, let's take the task of arranging the keys and break it into stages.

Let's call the 7 keys A, B, C, D, E, F and G

Stage 1: Place the A key on the ring.
We don't care about the "position" of the ring because all positions are the same. In other words, if I asked you "In how many ways can we place 1 key on a key ring?" the answer is 1.
So, we can complete stage 1 in 1 way.

Stage 2: Select a key to go to the immediate right of key A (i.e, clockwise from key A)
There are 6 remaining keys to digits from which to choose, so we can complete this stage in 6 ways.

Stage 3: Select a key to go to the immediate right of the key we selected in stage 2.
There are 5 keys remaining, so we can complete this stage in 5 ways.

Stage 4: Select a key to go to the immediate right of the key we selected in stage 3.
There are 4 keys remaining, so we can complete this stage in 4 ways.

Stage 5: Select a key to go to the immediate right of the key we selected in stage 4.
There are 3 keys remaining, so we can complete this stage in 3 ways.

Stage 6: Select a key to go to the immediate right of the key we selected in stage 5.
There are 2 keys remaining, so we can complete this stage in 2 ways.

Stage 7: Select a key to go to the immediate right of the key we selected in stage 6.
There is 1 key remaining, so we can complete this stage in 1 way.

By the Fundamental Counting Principle (FCP), we can complete all 7 stages (and thus arrange all 7 keys) in (1)(6)(5)(4)(3)(2)(1) ways (= 720 ways)

Answer: C

Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT, so be sure to learn this technique.

RELATED VIDEOS FROM OUR COURSE
Fundamental Counting Principle (FCP)



Fundamental Counting Principle - example


GMATPrepNow : There is a minor mistake (see highlighted part). Can you please update it to D.
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A key ring has 7 keys. How many different ways can the keys be arrange [#permalink]

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New post 23 Feb 2017, 21:37
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Bunuel wrote:
A key ring has 7 keys. How many different ways can the keys be arranged?

A. 6
B. 7
C. 120
D. 720
E. 5040


The circular arrangements of n distinct objects is represented by (n-1)! Because one of the n objects needs to be fixed.

Out of 7 different keys one needs to be fixed and remaining 6 keys can be arranged in 6! Ways
Hence, 6!=720 ways

Answer: Option D
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Last edited by GMATinsight on 23 Feb 2017, 23:51, edited 1 time in total.
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Re: A key ring has 7 keys. How many different ways can the keys be arrange [#permalink]

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New post 23 Feb 2017, 21:40
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GMATinsight wrote:
Bunuel wrote:
A key ring has 7 keys. How many different ways can the keys be arranged?

A. 6
B. 7
C. 120
D. 720
E. 5040


The circular arrangements of n distinct objects is represented by (n-1)! Because one of the n objects needs to be fixed.

Out of 7 different keys one needs to be fixed and remaining 6 keys can be arranged in 6! Ways
Hence, 6!=720 ways

Answer:Option C


GMATinsight : There is a minor mistake (see highlighted part). Can you please update it to D.
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Re: A key ring has 7 keys. How many different ways can the keys be arrange [#permalink]

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New post 23 Feb 2017, 23:54
0akshay0 wrote:
GMATinsight wrote:
Bunuel wrote:
A key ring has 7 keys. How many different ways can the keys be arranged?

A. 6
B. 7
C. 120
D. 720
E. 5040


The circular arrangements of n distinct objects is represented by (n-1)! Because one of the n objects needs to be fixed.

Out of 7 different keys one needs to be fixed and remaining 6 keys can be arranged in 6! Ways
Hence, 6!=720 ways

Answer:Option C


GMATinsight : There is a minor mistake (see highlighted part). Can you please update it to D.


Thank you!!! :)

Modified it.
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Re: A key ring has 7 keys. How many different ways can the keys be arrange [#permalink]

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New post 27 Feb 2017, 11:07
Bunuel wrote:
A key ring has 7 keys. How many different ways can the keys be arranged?

A. 6
B. 7
C. 120
D. 720
E. 5040


When reading this problem, we must notice that we are organizing the keys around a ring, in other words, a circle. Since we are arranging items around a circle, 7 keys can be arranged in (7-1)! = 6! = 720 ways.

Answer: D
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Re: A key ring has 7 keys. How many different ways can the keys be arrange   [#permalink] 27 Feb 2017, 11:07
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